A DCC-GARCH MODEL TO ESTIMATE

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1 8. A DCC-GARCH MODEL TO ESTIMATE THE RISK TO THE CAPITAL MARKET IN ROMANIA Marius ACATRINEI 1 Adrian GORUN 2 Nicu MARCU 3 Absrac In his paper we propose o sudy if he sandard and asymmeric dynamic condiional correlaion (DCC) models, following Cappiello e al. (2006), may capure spillover effecs and he degree of ineracion wih he European capial marke using he DAX index as proxy. We found evidence ha he asymmeric DCC models perform beer han he similar non-asymmeric ones. In he second semeser of 2011, increased significan dynamic correlaions sugges he presence of volailiy spillovers from he main capial equiy markes. Alhough all DCC models can capure conagion, seen as a significan increase in he co-movemens of sock index reurns, he AGD-DCC model is more sensiive o unexpeced changes in reurns. The resuls indicae significan, bu no very srong correlaion of BET and BETFI indexes wih he DAX index in he second semeser of Keywords: volailiy spillovers, conagion effecs, sock reurn comovemens JEL Classificaion: G01, G14, G32 I. Inroducion While he univariae GARCH models can analyze he variance of he marke shock in he univariae asymmeric models, he recen developmen of he class of mulivariae GARCH models led o incorporaion of he asymmeric response of reurns o he marke shocks. 1 Insiue for Economic Forecasing, Romanian Academy; marius.acarinei@gmail.com 2 Consanin Brancusi Universiy of Targu Jiu 3 Universiy of Craiova; marcu.nicu@yahoo.com 136 Romanian Journal of Economic Forecasing 1/2013

2 A DCC-GARCH Model o Esimae he Risk o he Capial Marke in Romania The mulivariae GARCH models are usually used o analyze he volailiies and covolailiies across markes (Kearney and Paon, 2000), being designed o quanify he way in which he news is influencing he marke volailiies. Cappiello, Engle and Sheppard (2006) designed an asymmeric version of he DCC model of Engle (2002) in order o examine he degree in which changes in asse correlaion show evidence of asymmeric responses o negaive reurns. Examining he correlaion beween he Romanian bes known indexes, namely he BET and he BET-FI and he DAX index, could supply a heoreical jusificaion for invesors who seek o hedge porfolio exposures. Since correlaion is ime-varying, we believe ha a shor-run analysis should poin ou he degree o which, for insance, he reurns volailiy increase in a bear marke in co-movemen wih oher volailiies. Good esimaes of he covariance marix and correlaion srucure of he reurns are very imporan for a porfolio manager or for a risk manager. This is he reason why we chose o analyze he shor-run reurns in 2011, when he European sovereigny deb crisis had spread o all classes of asses. We are ineresed o see if he financial uncerainy due o he Greek crisis and low growh environmen forecased for he oher European counries in 2011 led o increased correlaion among he asses on he Romanian capial marke and, if so, wheher an asymmeric DCC-GARCH model migh supply a beer fi, and wheher he co-movemens in he sock reurns were associaed wih he spread of conagion. The findings will also help us derive conclusions abou invesors keenness o move capial o he Easern Europe or wheher he Romanian capial marke is decoupled from he European capial markes. The paper is srucured as follows: secion 2 presens a review of he recen lieraure for mulivariae GARCH models and developmens relaed o he DCC models, and in secion 3 we presen he economeric mehodology. In secion 4 we describe he daa used in he paper, while in secion 5 we presen he resuls. Secion 6 concludes and discusses areas for furher research. II. Lieraure Review Mulivariae GARCH models involve he esimaion of he covariance marix which can be made eiher direcly, as in he VEC (Bollerslev e al., 1988), DVEC, and diagonal VEC - a resriced version of VEC, or BEKK (Engle e al., 1995) models or indirecly, using condiional correlaions as in CCC, DCC or STCC (Smooh Transiion Condiional Correlaions) models. The VEC, DVEC and BEKK models face he problem of esimaing a very large number of parameers. The orhogonal GARCH (O-GARCH) proposed by Alexander (2000) represens he errors as linear combinaions of uncorrelaed facors (similar o principal componen analysis), in a smaller number han he error vecor in order o reduce he dimension of he covariance marix. In a highly correlaed sysem, only a few principal componens are necessary o describe adequaely he reurns volailiy, and large porfolios may be hus esimaed. If he reurns are weakly correlaed, or he componens have similar uncondiional variance, problems in he esimaion of O- GARCH will occur. Romanian Journal of Economic Forecasing 1/

3 Insiue for Economic Forecasing Van der Weide (2002) generalized he O-GARCH model as a GO-GARCH model in order o solve some of he problems. The issue of maximizing he MLE funcion for larger porfolios led o he developmen of a hree-sep procedure (Boswijk and van der Weide, 2006). A differen procedure for he esimaion of he GO-GARCH model was developed by Broda and Paolella (2008) by using he independen componen analysis (ICA) o reduce he dimension problem o a se of univariae models. The mehod is called CHICAGO (Condiionally Heeroskedasic Independen Componen Analysis of GO- GARCH) and allows for non-normal innovaions. Moreover, in pracice i is requesed o develop models ha ake ino accoun mulivariae issues such as volailiy spillover and conagion effecs across markes. The GARCH models are sill widely considered models for measuring he financial risk. The ineres shown in he class of DCC models is given by he fac ha i calculaes he correlaion beween he asse reurns as a funcion of heir pas volailiy and he correlaions among hem. Coming from he GARCH family, a DCC model uses he recen pas informaion for esimaing he presen correlaion beween series. Since correlaion may be measured as well as volailiy, he esimaion of a DCC model is usually achieved in wo seps in order o simplify he esimaion of he ime varying correlaion marix. The DCC model was inroduced by Engle (2002) and is specificaions will be discussed in he nex secion. In comparison wih oher correlaion models, among which scalar BEKK, diagonal BEKK, O-GARCH, he DCC wih inegraed moving average esimaion, he DCC by log likelihood for inegraed model and he DCC by log likelihood for mean revering model, he las one scored beer han he ohers. The selecion of he bes model was made by using several ess, such as he mean absolue error es, auocorrelaion es of he squared sandardized residuals and an esimaor of he value of risk for wo-asse porfolio (Engle, 2002). Similar dynamic correlaion models were shorly developed aferwards by Chrisodoulakis and Sachell (2002) and by Tse and Tsui (2002) he TVC (Time Varying Correlaion) model. A differen specificaion of he H is given in he correced DCC (cdcc) model of Aielli (2009). There is empirical evidence ha in pracice boh models have close resuls. Anyway, by allowing he sandard DCC model o incorporae asymmeries (Cappiello e al., 2006) beer resuls were obained by modeling he condiional correlaions. The ime varying dependence across asses is he copula-garch or, more recenly, copula-vine approach. The join disribuion funcion may be decomposed ino N marginal disribuions and a copula funcion which describes he dependence beween he N asses (see Jondeau (2006) for a copula-garch model). Several dynamic copula-garch models, which assume ha he copula parameers evolve according o a ime-varying condiional correlaion marix, were applied o he Romanian, Bulgarian, Polish and Czech sock index reurns and he models fied wih skew- residuals showed beer resuls han he Gaussian or -residuals models (Acarinei, 2011). If he margins are normal and he copula is mulivariae normal, hen he dependence is described by he correlaion marix. If no, hen he assumpion of mulivariae normal disribuion is no realisic for modeling asse reurns and a copula approach should be used insead. 138 Romanian Journal of Economic Forecasing 1/2013

4 A DCC-GARCH Model o Esimae he Risk o he Capial Marke in Romania Since in pracice here are many esimaion issues, such as he esimaion of randomly chosen subsample, may produce differen correlaions, or he bivariae esimaion, as recommended by many researchers, may give differen parameers for correlaion, or he dynamics of he reurns may have differen regimes, he laes models developed for condiional correlaions include: a quadraic flexible DCC model (Billio and Caporin, 2009), a generalized DCC model (Hafner and Franses, 2009), a regime swiching DCC (Pelleier, 2006), a componen DCC (Colacio e al., 2009), STCC model (Silvennoinen and Terasvira, 2005), a facor-spline-garch (Rangel and Engle, 2009), dynamic copula-garch and copula-vine models (Aas e al.,2009). Deailed surveys of Mulivariae GARCH models are given in Bauwens e al. (2006) and Silvennoinen and Terasvira (2009). III. Dynamic Condiional Correlaion Models The sudy of Cappiello e al. (2006) uses a generalizaion of he sandard DCC model inroduced by Engle (2002) and includes he asse-specific correlaion of news impac curves and he asymmeric dynamics in correlaion. The asse reurns, r, are condiionally normal wih mean zero, which is a sylized fac, and he condiional covariance marix, H. Following Engle and Sheppard (2001), he condiional covariance marix can be decomposed as: H = D R D where: D is a kxk diagonal marix of ime-varying sandard deviaions from univariae GARCH models and R is he ime-varying correlaion marix. The mulivariae normal disribuion was iniially assumed in he sandard DCC model, bu we may model he reurns wih oher disribuions. The key elemen is he correlaion marix,, which is ime varying in comparison wih he Consan Condiional Correlaion model (CCC) in which he correlaion is assumed consan, namely H = D RD. The likelihood of he DCC esimaor is: The volailiy ( D ) and he correlaion ( R ) componens may vary, hus he esimaion process is achieved in wo seps. Firsly he volailiy (L v ) is maximized: T ' L = klog π + log D + r D 2 r hen he correlaion (L c ) is maximized: ( ( ) ( ) ) v = 1 T ' 1 ' ( ( ) ε ε ε ε ) L = 0.5 log R + R c = 1 (See Engle and Sheppard (2001) for he esimaion of he log-likelihood funcion). Romanian Journal of Economic Forecasing 1/

5 Insiue for Economic Forecasing Since he number of he parameers o be esimaed is (n+1) (n+4)/2 large, Engle proposed a wo-sep esimaion. The DCC model is esimaed by a wo-sep procedure: a) in he firs sep univariae GARCH models are fied for each asses reurns and esimaes of heir variances are hus obained; b) he reurns are filered ou of he GARCH effec (degarched reurns) by dividing by heir esimaed sandard deviaions and hen are used o esimae he dynamics of correlaion, εi = r i / hi. In he second sep, he sandardized residuals are used o esimae he ime-varying correlaion marix. The model developed by Engle (2002) has he following non-linear GARCH specificaion for he condiional correlaion: ' Q = ( 1 a b) Q+ aε 1ε 1+ bq 1 where Q = ( qij ) is a nxn symmeric posiive definie marix, a and b are nonnegaive scalars such as a+ b< 1, a is he news coefficien and b is he decay ' coefficien. Q = E εε is he uncondiional variance marix of he sandardized residuals (he uncondiional correlaion). The condiional correlaions q ij are imevarying and follow a srucure similar o a GARCH (1, 1) model. Engle showed ha modeling he condiional covariance of he sandardized reurns is equivalen o modeling he condiional correlaion of he reurns. For ensuring a condiional correlaion beween -1 and +1, by normalizaion he correlaion can be expressed as ρ ij, = qij, / qii, qjj,. As in he case of GARCH models, if a+ b< 1, he model is mean-revering, and if he sum of he parameers is equal o 1, hen he model is inegraed. The correlaions are obained by ransforming his o: R = ( diagq) Q( diagq) where ( diagq ) 0.5 is a diagonal marix of he square roo of he diagonal elemens of Q. The limiaion of he sandard DCC model is he assumpion ha he condiional correlaions follow he same dynamic srucure, in conras o he Markov Swiching Model or a Threshold Auoregressive Model where differen dynamics may be assumed. If he daa have srucural breaks, he condiional correlaion models may lead o incorrec esimaion of he risk. Also, he DCC model is limied o a small number of asses. A GO-GARCH model could simplify compuaional requiremens for large porfolios. In order o capure he asymmeries in he daa, differen asymmeric mulivariae GARCH models were developed (Cappiello e al., 2006). The univariae volailiy models were seleced by he Schwarz Informaion Crierion (BIC) from he GARCH family capable of capuring he sylized facs of asse reurns. 140 Romanian Journal of Economic Forecasing 1/2013

6 A DCC-GARCH Model o Esimae he Risk o he Capial Marke in Romania In his respec, we used he following asymmeric models ha capure he leverage effec in a differen way: he EGARCH model (Nelson, 1991) and he GJR-GARCH model (Glosen, Jaganahan and Runkle, 1993), since GJR and EGARCH allow for hreshold effecs bu use differen powers of variance in he variance equaion. For each GJR and EGARCH model we modeled he mean equaion wih an AR(1) and ARMA(1,1) specificaion. Because he sandard DCC developed by Engle does no include asymmeries, he equaion was modified in order o incorporae he asymmeries and asse-specific impac parameers of news. ' ( ) ε ε Q = Q AQA BQB GNG + A' A + B ' Q B + G ' n n ' G n = I ε < ε and where A,B,G are diagonal parameer marices, [ 0] N E[ nn' ] =. Thus i is compued he Q marix a ime, given he firs lag of Q and he sandardized residuals. We may see ha here are four versions of he model: 0,, 1) he DCC Garch model if [ ] G = A= a = a ij ij B = b = b ; 2) he Asymmeric DCC Garch model (ADCC) if ij B = b = b ij, ij G = g = g ; 3) he Generalized Diagonal DCC Garch model (GDDCC) if G = [ 0] ; A= a = a, 4) he Asymmeric Generalized Diagonal DCC Garch (AGDDCC) model was developed o capure he heerogeneiy in reurns, so ha i allows for differen news impac and smoohing parameers across he asses (for more informaion see Cappiello, Engle and Sheppard, 2006). IV. Daa The daa used in his paper are he daily closing prices of wo Romanian sock exchange indexes, namely BET and BET-FI, and he DAX index in The daa concerning he Romanian indexes are available from he Buchares Sock Exchange websie and he daa concerning DAX are available a he yahoo websie. All daa are denominaed in euro. We did no use pseudo-closes, namely sampling he prices a he same GMT, bu he closing prices. There are 250 observaions from January 4, 2011 o December 23, Considering heir properies, we may see ha he daa have he properies usually noiced in he case of financial reurns: he reurns are lepokuric, have negaive Romanian Journal of Economic Forecasing 1/

7 Insiue for Economic Forecasing skew, and exreme excess kurosis. Generally, by sandardizing he reurns, hey can be normal or close o normal. To invesigae he properies of innovaions, we sandardized he residuals in every GARCH model. The residuals obained were less skewed and less fa-ailed, bu hey were sill non-normal, rejecing he Jarque-Bera es a 1% level. Therefore, he univariae GARCH models applied o he sock index reurns were modeled wih a Suden- disribuion. V. Empirical Resuls We modeled each ime series wih an EGARCH and GJR model, using a Suden- disribuion, while for he mean equaion we used an AR(1) and ARMA(1,1) specificaion. Table 1 Log Likelihood of he Esimaed Models AR(1)- EGARCH(1,1)- ARMA (1,1)- EGARCH(1,1)- AR(1)- GJR(1,1)- ARMA(1,1)- GJR(1,1)- BETFI BET DAX Table 1 presens he log likelihood of he univariae GARCH models esimaed for BETFI, BET and DAX sock index reurns. We see ha, alhough he models are differen in specificaions, hey all come very close, wih a lile improvemen for he AR(1)-GJR(1,1) wih Suden- disribuion. We esimaed he same models wih he normal disribuion, bu he models esimaed wih he Suden- disribuion showed significanly beer resuls. We have an expecaion ha he reurns should show some significan correlaion in Augus and November, 2011 since a lo of evens happened a ha ime. We menion only some which occurred in Augus 2011: some of he bigges drops in sock prices in he USA, Europe and Asia due o fear of conagion of he European deb crisis owards Spain and Ialy in he firs place. A reform of he Spanish consiuion was necessary in Augus for winning back marke confidence. S&P downgraded America' credi raing from riple A o AA+. Many imporan sock exchanges faced severe declines: CAC40 fell by 20% in wo weeks, he DAX fell by 5.8% in one day on 18 Augus and FTSE100 fell by 4.5% also on 18 Augus. Also, widespread fears abou he reliabiliy of he Greeks banks were ever-presen and alks abou he sabiliy of he Euro Zone led o a higher volailiy across he sock markes. Figures 1-3 depic four models for dynamic correlaions beween BETFI and BET, BET and DAX, BETFI and DAX. The sock index reurns were degarched using only he AR(1)-EGARCH(1,1)- and AR(1)-GJR(1,1)- models, since some of he ARMA coefficiens were no saisically significan. 142 Romanian Journal of Economic Forecasing 1/2013

8 A DCC-GARCH Model o Esimae he Risk o he Capial Marke in Romania Figure 1 Dynamic correlaions beween BETFI and DAX AR(1)-EGARCH(1,1)- AR(1)-GJR(1,1)- Figure 2 Dynamic correlaions beween BET and DAX AR(1)-EGARCH(1,1)- AR(1)-GJR(1,1)- The dynamic correlaions of DCC and A-DCC models are almos he same for he EGARCH model, while he AGD-DCC (GJR) model shows a differen dynamics, wih greaer ampliude in comparison wih DCC, A-DCC and GD-DCC, which follow a Romanian Journal of Economic Forecasing 1/

9 Insiue for Economic Forecasing similar paern. The GD-DCC and AGD-DCC show a similar paern when using he EGARCH specificaion. All models show a significan correlaion beween BETFI and BET, wihin he [0.5, 0.9] inerval. The highes correlaion was reached in Augus, hus showing ha he AGD-DCC model was more sensiive o incorporae negaive news han he oher models. The correlaion beween BET and DAX becomes significan in he second semeser, wih he momenum of he European crisis for boh specificaions. We may noice a clear spike in he GD-DCC model wih EGARCH specificaion in Augus. The AGDDCC and GD-DCC are more volaile han he ohers. The GD-DCC and A-DCC models, wih GJR specificaion, capure he spillover from Greek evens in November. Figure 3 Dynamic correlaions beween BETFI and DAX AR(1)-EGARCH(1,1)- AR(1)-GJR(1,1)- We see he same seesaw paern as before for he GD-DCC and AGD-DCC models. There are spikes of correlaion ha are significan, over 0.5, corresponding mainly o he influence of exernal facors such as he escalaion of he Greek crisis. We decided o answer he quesion which he bes DCC-GARCH model is o esimae he Romanian capial marke risk seen in inerdependence wih oher capial markes. According o he BIC informaion crieria, we seleced individual GARCH models which capure leverage in he variance equaion and wih auoregressive and moving average erms for he mean equaion. The bes model seems o be he AGD-DCC, an asymmeric generalized diagonal DCC model ha does include asymmeries. Very close o i is he GD-DCC model. We see in he above figures ha boh models are capable of capuring spillovers from oher capial markes, while he oher models, alhough close o hem, are no so responsive. 144 Romanian Journal of Economic Forecasing 1/2013

10 A DCC-GARCH Model o Esimae he Risk o he Capial Marke in Romania Our inuiion was ha he condiional equiy correlaion is bound o increase when bad news affec he financial markes. For his reason, he class of asymmeric models should provide a beer model for he condiional correlaion. I remains o es in oher paper how hey respond o he posiive news and wheher hey produce beer forecass han he non-asymmeric models. Therefore, we should see if he asymmeric models suffer from a lack of addiional effeciveness because of poenial misspecificaion of he univariae GARCH models employed or o accumulaion of esimaion errors because of a larger number of he model parameers. Table 2 Log-likelihood for he DCC Esimaed Models Log-likelihood AR(1)-EGARCH(1,1)- AR (1)-GJR(1,1)- DCC A-DCC GD-DCC AGD-DCC The peak of he European sovereign deb crisis and he bigges drops in equiy across European sock markes can be locaed in Augus, while he oher spell of uncerainy was in November All dynamic correlaions for BETFI and BET show ha heir highes correlaion occurred in Augus, indicaing ha he main indexes of he Romanian capial marke responded o exernal evens and ha he AGD-DCC models can incorporae volailiy spillovers. The resuls indicae ha here is a significan increase in he co-movemens of sock index reurns, hus indicaing he spread of conagion on he Romanian capial marke. Figure 4 An AGD-DCC model wih EGARCH/GJR specificaions Romanian Journal of Economic Forecasing 1/

11 Insiue for Economic Forecasing Oher ess may prove useful in order o deermine false periods of significan correlaion. Figure 4 shows how differen he resuls may be, even if we use close specificaion for GARCH univariae processes. The AGD-DCC model wih GJR specificaion has a seesaw paern, wih correlaions becoming significan only in he second half of The oher model has a more seady dynamics, implying a more sable dynamics beween BET and DAX indexes, alhough insignifican in he firs semeser, bu converging owards he dynamics of he AGD-DCC (GJR) model. VI. Conclusions We inended o sudy asymmeric DCC-GARCH models capable of idenifying volailiy spillover and conagion effecs across capial equiy markes. The curren inernaional financial urmoil revealed a high inerdependence beween he capial equiy markes, as high volailiies were recorded simulaneously on he inernaional sock markes. In his paper, we invesigaed if here are any volailiy spillovers from developed capial markes and conagion effecs, namely beween he Romanian capial marke and he European capial marke, aking he German index (DAX) as a proxy, given is imporance for oher financial markes. In his respec, we used he daily reurns of he main sock indexes of hese markes, BET and BETFI for he Romanian capial marke and he DAX index, observed in 2011, in order o invesigae he shor-run dynamics correlaion beween hem. Following Cappiello e al. (2006) we employed four DCC models, namely he DCC Garch model (DCC), he Asymmeric DCC Garch model (A-DCC), he Generalized Diagonal DCC Garch model(gd-dcc), and he Asymmeric Generalized Diagonal DCC Garch (AGD-DCC) model. Ou of he four models, wo were asymmeric. There is evidence ha he AGD-DCC model is more sensiive o negaive news han he oher DCC models, while having he bes fi irrespecive of he GARCH specificaion used, ha is AR(1)-EGARCH(1,1) and AR(1)-GJR(1,1) wih Suden- disribuion. Oher GARCH specificaions should also be esed. We noed ha he condiional correlaions of he BET index and he DAX index considerably increased during he crisis period, namely in he second semeser of 2011, when he European deb sovereign crisis reached is peak. There is evidence o conclude ha during he crisis he volailiy spillovers and conagion effecs were significan, bu no very srong beween he Romanian and he German capial markes. Therefore, we may say ha he Romanian capial marke responds o some exen o exernal influences. I remains o be esed in oher paper how indices respond o posiive news and wheher he asymmeric DCC models produce beer forecass han he non-asymmeric models. The model resuls agree wih he conclusions of Cappiello e al. (2006), namely ha he log-likelihood of he models increases when we include asymmeric effecs. A las, we would like o sugges ha sudying dynamic condiional correlaions beween markes is a pracical endeavor, a many differen levels, from developing Value a Risk esimaion for porfolio managers who need o hedge heir porfolio exposure o financial risk, o designing beer risk indicaor ools for risk manager 146 Romanian Journal of Economic Forecasing 1/2013

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